# A fast algorithm for maximum likelihood estimation of mixture proportions using sequential quadratic programming

arXiv: https://arxiv.org/abs/1806.01412 · HTML: https://mlsanity.com/paper/1806.01412/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2018.

## Built on by (most cited first, 4 of 4)

- [A flexible empirical Bayes approach to multiple linear regression and connections with penalized regression](https://mlsanity.com/paper/2208.10910/) (2022)
- [A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regression](https://mlsanity.com/paper/2309.16843/) (2023)
- [Nonparametric MLE for Gaussian Location Mixtures: Certified Computation and Generic Behavior](https://mlsanity.com/paper/2503.20193/) (2025)
- [Nash: Neural Adaptive Shrinkage for Structured High-Dimensional Regression](https://mlsanity.com/paper/2505.11143/) (2025)

## Abstract (arXiv)

Maximum likelihood estimation of mixture proportions has a long history, and continues to play an important role in modern statistics, including in development of nonparametric empirical Bayes methods. Maximum likelihood of mixture proportions has traditionally been solved using the expectation maximization (EM) algorithm, but recent work by Koenker & Mizera shows that modern convex optimization techniques -- in particular, interior point methods -- are substantially faster and more accurate than EM. Here, we develop a new solution based on sequential quadratic programming (SQP). It is substantially faster than the interior point method, and just as accurate. Our approach combines several ideas: first, it solves a reformulation of the original problem; second, it uses an SQP approach to make the best use of the expensive gradient and Hessian computations; third, the SQP iterations are implemented using an active set method to exploit the sparse nature of the quadratic subproblems; fourth, it uses accurate low-rank approximations for more efficient gradient and Hessian computations. We illustrate the benefits of our approach in experiments on synthetic data sets as well as a large genetic association data set. In large data sets (n = 1,000,000 observations, m = 1,000 mixture components), our implementation achieves at least 100-fold reduction in runtime compared with a state-of-the-art interior point solver. Our methods are implemented in Julia, and in an R package available on CRAN (see https://CRAN.R-project.org/package=mixsqp).